Equation, Butler-Volmer Nernst

For the electrode plane ( = 0), two separate boundary conditions are required as the plane is composed of two different materials the electroactive microdisc and the insulating supporting surface. As the microdisc surface is electroactive, a potential-dependent boundary condition is applied the Nernst equation, Butler-Volmer or Marcus-Hush models may be used as appropriate. 

Guoy-Chapman equation [ ] Nernst equation [ ] Tafel equation [ ] Butler-Volmer equation... 

This general equation covers charge transfer at electrified interfaces under conditions both of zero excess field, low excess fields, and high excess fields, and of the corresponding overpotentials. Thus the Butler-Volmer equation spans a large range of potentials. At equilibrium, it settles down into the Nernst equation. Near equilibrium it reduces to a linear / vs. T) (Ohm slaw for interfaces), whereas, if T) > (RT/fiF) (i.e., one is 50 mV or more from equilibrium at room temperature), it becomes an exponential /vs. T) relation, the logarithmic version ofwhich is called Tafel s equation. 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

Derive the Nernst equation from the Butler-Volmer equation. 

For the reversible case, the Nernst equation applies instead of the Butler-Volmer equation, that is, in dimensionless terms as in (2.32), rewritten as... 

We have the following unknown boundary values the two species nearsurface concentrations Cyo and Cb,o, the two species fluxes, respectively G and G n, the additional capacitive flux Gc, and the potential p, differing (for p > 0) from the nominal, desired potential pnom that was set, for example, in an LSV sweep or a potential step experiment. Five of the six required equations are common to all types of experiments, but the sixth (here, the first one given below) depends on the reaction. That might be a reversible reaction, in which case a form of the Nernst equation must he invoked, or a quasi-reversible reaction, in which case the Butler-Volmer equation is used (see Chap. 6 for these). Let us now assume an LSV sweep, the case of most interest in this context. The unknowns are all written as future values with apostrophes, because they must, in what follows below, be distinguished from their present counterparts, all known. 

In spite of the above justification for the kinetic approach to the estimate of l, this has a number of drawbacks. First of all, there is no point in using a kinetic approach to determine a thermodynamic equilibrium quantity such as l. The justification of the validity ofEqs. (42) and (45) by the resulting equilibrium condition of Eq. (46) is far from rigorous, just as is the justification of the empirical Butler-Volmer equation by the thermodynamic Nernst equation. Moreover, the kinetic expressions of Eq. (41) involve a number of arbitrary assumptions. Thus, considering the adsorption step of Eq. (38a) in quasi-equilibrium under kinetic conditions cannot be taken for granted a heterogeneous chemical step, such as a deformation of the solvation shell of the... 

Nernst or Butler-Volmer equation (Neumann boundary) Used to define the concentration ratio at the electrode surface when electrolysis is not transport limited. 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

In the given form, the Butler-Volmer equation is applicable rather broadly, for flat model electrodes, as well as for heterogeneous fuel cell electrodes. In the latter case, concentrations in Eq. (2.13) are local concentrations, established by mass transport and reaction in the random composite structure. At equilibrium,/f = 0, concentrations are uniform. These externally controlled equilibrium concentrations serve as the reference (superscript ref) for defining the equilibrium electrode potential via the Nernst equation. 

Note that this does not mean that the concentration profile of species A is equivalent to that of the E mechanism since it will be influenced by the chemical reaction through the surface boundary conditions. Thus, the chemical reaction affects the surface concentration of species B, which is related to that of species A through the Nernst equation (for reversible systems), or more generally, through the Butler-Volmer or Marcus-Hush relationships. Therefore, the surface concentration of species A, and as a consequence the whole concentration profile, will reflect the presence of the chemical process. 

The Warburg and Nernst impedances were derived under the assumption that the potential obeys the Nernst equation. The more realistic Randles model takes into account the kinetics of charge transfer as described by the Butler-Volmer equation. For the electrode reaction (5.147) this is written as... 

At equilibrium the Butler-Volmer equation should transform into the Nernst equation. This condition gives the activation energy in Eq. 

Analysis becomes more complex when the the interfacial electron transfer kinetics at the support electrode/film interface are slow. In such a situation the full Butler-Volmer equation must be used instead of the Nernst equation. In this more complex case, a new variable A characterizing the degree of reversibility of the kinetics at the inner interface is introduced, where A is given by... 

This is the famous Butler-Volmer equation. Incorporating the Nernst equation, which relates the equilibrium electrode potential to the standard equilibrium potential and to the equilibrium composition of the bulk electrolyte (concentrations with superscript b) via... 

The model of water-filled nanopores, presented in the section ORR in Water-Filled Nanopores Electrostatic Effects in Chapter 3, was adopted to calculate the agglomerate effectiveness factor. As a reminder, this model establishes the relation between metal-phase potential and faradaic current density at pore walls using Poisson-Nernst-Planck theory. Pick s law of diffusion, and Butler-Volmer equation... 

The Butler-Volmer Equation and the Nernst Equation Problem... 

Show how the Butler-Volmer equation (as given below) reduces to the Nernst equation for a reversible one-electron process ... 

Two boundary conditions are required for each species one in bulk and one at the electrode. Conventionally the concentrations are set to their initial values in bulk, and at the electrode either the Nernst equation or Butler-Volmer equation is applied to describe the electrode kinetics. These equations have the general form/(c,o E) = 0, where the applied potential is a linear function of time. Conservation of mass also requires that the fluxes of reactant A and product B are equal and opposite at the electrode surface. 

The wave shapes observed for electrochemically irreversible or quasi-reversible voltammograms are governed by Pick s law of diffusion (Eq. II. 1.6) and the Butler-Volmer expression (Eq. II.l. 16). By rewriting the Butler-Volmer equation for the case of a reduction A -I- ne B (Eq. 11.1.19), it can be shown that, for the limit of extremely fast electron transfer kinetics, the Nernst law... 

Figures 9 and 13 show current potential curves for reversible and steady state processes. The curves were calculated from the Nernst and Butler-Volmer equations, respectively. If an experimental curve is adequately described by the Nernst equation, this is a reversible reaction. However, it is not convenient to check the current potential curve itself for reversibility, it is much easier to replot it according to equation 3.43. The potential is plotted on the abscissa and log on the ordinate. If a straight line is...

Once the local concentration overpotential is known, the activation overpotential, ria, is obtained by subtracting Tjc from total Tj. The local activation overpotential is the actual driving force of the electrochemical reaction. It is related to the local current density at any point of the reaction zone by an electrochemical rate equation such as the Butler-Volmer equation (Eq. (10a)). Therefore, the rate equation, the Nernst equation (Eq. (37)), and the potential balance in combination couple the electric field with the species diffusion field. In addition, the energy balance applies also at the electrode level. Although this introduces another complication, a model including a temperature profile in the electrode is very useful because heat generation occurs mainly by electrochemical reaction and is localised at the reaction zone, while the... 

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